Deriving Survival Probability from a Time-Varying Default Hazard

Summary

The document explains how to solve the differential equation for a firm’s survival probability when the default hazard rate varies over time. It begins with the condition that the firm has not defaulted at the initial time, so survival probability starts at one. Dividing the differential equation by the survival probability and integrating from the initial time to the horizon turns the left side into the change in its logarithm.

Exponentiating the resulting relation gives survival probability as the exponential of the negative cumulative hazard. The integration variable can be written separately from the endpoint to clarify that the hazard is accumulated across the interval. This is a concise derivation of a standard relationship used in credit risk. The explanation assumes the stated hazard-rate model and initial survival condition; it does not discuss how hazard rates are estimated, whether default intensity is stochastic, or how survival probabilities translate into credit prices or recovery-adjusted default probabilities.

Key ideas

• Survival probability starts at one when default at the initial time is ruled out.
• Dividing the differential equation by survival probability makes its left side a logarithmic derivative.
• Integrating the hazard rate over time gives cumulative hazard.
• Exponentiating the negative cumulative hazard yields the survival probability at the horizon.
• The derivation does not explain hazard-rate estimation or credit instrument valuation.

Tags

# Help with simple derivation of probability of credit default


# Help with simple derivation of probability of credit default












I'm going over a chapter in Hull's Options, Futures, and Other Derivatives and am stuck on how the probability of default is derived. Here's the image of the derivation.

I can follow all of it except for one step: how do you derive $V(t) = e^{-\int_0^t \lambda(\tau) \,d\tau}$ from $\frac{dV(t)}{dt} = -\lambda (t)V(t) $ ?

I'm not a quant so I don't really know how to proceed. I can just plug in the formula in my project, but I'd rather understand how/why the derivation works.

## Answer by Pleb (score 2, accepted)

https://quant.stackexchange.com/a/70420

First and foremost, assume that the company can not default at time $t=0$, implying that $V(0)=1$.

Now, divide with $V(t)$ on both sides and integrate from 0 to $t$:

$$ \int_0^t \frac{\frac{dV(t)}{dt}}{V(t)} dt = - \int_0^t \lambda(t) dt $$ Calculate the LHS: \begin{align} \ln(V(t)) - \ln(V(0)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ \ln(V(t)) &= - \int_0^t \lambda(t) dt\\ &\Updownarrow\\ V(t) &= e^{ - \int_0^t \lambda(t) dt}, \end{align} where you can substitute $t$ with $\tau$ in the integrand on the RHS (the hazard rate function) in order to alleviate notational confusion. This will give you the desired result.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.